A train in standing on a platform, a man inside a compartment of a train drops a stone. At the same instant train starts to move with constant acceleration. The path of the particle as seen by the person who drops the stone is:

Options

A. parabola
B. straight line for some time & parabola for the remaining time
C. straight line
D. variable path that cannot be defined

Explanation

To solve the problem, we need to analyze the motion of the stone dropped by the man inside the moving train. The train starts moving with a constant acceleration at the same moment the stone is dropped.

### Step-by-Step Solution:

1. **Identify the Reference Frames**:
- The man inside the train is in a non-inertial frame because the train is accelerating.
- The stone is dropped from rest relative to the man.

2. **Motion of the Stone**:
- When the stone is dropped, it has an initial velocity of 0 m/s in the vertical direction (downward).
- The only force acting on the stone after it is dropped is gravity, which causes it to accelerate downward with an acceleration \( g \).

3. **Motion of the Train**:
- The train starts moving with a constant acceleration \( a \) to the right.
- As the train accelerates, the position of the man in the train changes relative to the stone.

4. **Equations of Motion**:
- For the stone (in the vertical direction):
\[ y = \frac{1}{2} g t^2 \]
- For the train (in the horizontal direction):
\[ x = \frac{1}{2} a t^2 \]

5. **Eliminate Time (t)**:
- From the equation of motion for the train, we can express time \( t \) in terms of \( x \):
\[ t = \sqrt{\frac{2x}{a}} \]
- Substitute this expression for \( t \) into the equation for the stone:
\[ y = \frac{1}{2} g \left(\sqrt{\frac{2x}{a}}\right)^2 \]
- Simplifying this gives:
\[ y = \frac{1}{2} g \cdot \frac{2x}{a} = \frac{g}{a} x \]

6. **Conclusion**:
- The equation \( y = \frac{g}{a} x \) represents a straight line passing through the origin.
- Therefore, the path of the stone as seen by the man in the train is a straight line.

### Final Answer:
The path of the particle (stone) as seen by the person who drops it is a straight line.